cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A371227 E.g.f. satisfies A(x) = 1 - x*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 56, 390, 6384, 92400, 1812768, 38565072, 949927680, 25934040000, 783458550720, 25909868761920, 930720395219328, 36108805836317760, 1504050682102456320, 66964478742976711680, 3173178938051223889920, 159461567895099436047360
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n-2*k)!*abs(stirling(n-k, k, 1))/((n-k)!*(2*n-3*k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*n-2*k)! * |Stirling1(n-k,k)|/( (n-k)! * (2*n-3*k+1)! ).

A371229 E.g.f. satisfies A(x) = 1 - x*A(x)^2*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 104, 630, 19944, 259560, 8718464, 185086944, 6914815200, 206059083120, 8700740615808, 332779651158240, 15916427365716864, 738672634596405600, 39847940942657495040, 2163098542598925281280, 130682368989193123952640
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*(2*n)!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/((n-k)!*(2*n-k+1)!));

Formula

a(n) = n! * (2*n)! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/( (n-k)! * (2*n-k+1)! ).

A371271 E.g.f. satisfies A(x) = 1 + x*A(x)^3 * (exp(x*A(x)^2) - 1).

Original entry on oeis.org

1, 0, 2, 3, 124, 725, 28146, 352807, 14395256, 298559529, 13269150190, 394087597211, 19361289265044, 752705527798237, 41083484117561354, 1970818974867113295, 119467697774656366576, 6792102349650727753553, 455778318504089893121766
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n+k)!*stirling(n-k, k, 2)/(n-k)!)/(2*n+1)!;

Formula

a(n) = (n!/(2*n+1)!) * Sum_{k=0..floor(n/2)} (2*n+k)! * Stirling2(n-k,k)/(n-k)!.

A371228 E.g.f. satisfies A(x) = 1 - x*A(x)*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 80, 510, 12084, 164640, 4272736, 91935648, 2769703920, 80692896240, 2849745645504, 103479044628960, 4250475820200960, 183436357950387360, 8649275730513361920, 430735131434242736640, 22999938416454315239424, 1295673669960473064844800
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (2*n-k)!*abs(stirling(n-k, k, 1))/((n-k)!*(2*n-2*k+1)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (2*n-k)! * |Stirling1(n-k,k)|/( (n-k)! * (2*n-2*k+1)! ).

A371232 E.g.f. satisfies A(x) = 1 - x*A(x)^4*log(1 - x*A(x)^3).

Original entry on oeis.org

1, 0, 2, 3, 176, 1050, 57144, 744660, 41682304, 917959392, 54654865920, 1761420386880, 113338947830976, 4879197834619680, 341937322823859840, 18486700938579444480, 1415296984669095859200, 92017658919053166405120, 7695907229874069158658048
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, (3*n+k)!*abs(stirling(n-k, k, 1))/(n-k)!)/(3*n+1)!;

Formula

a(n) = (n!/(3*n+1)!) * Sum_{k=0..floor(n/2)} (3*n+k)! * |Stirling1(n-k,k)|/(n-k)!.

A371235 E.g.f. satisfies A(x) = 1 - x^2*A(x)^5*log(1 - x*A(x)^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 5220, 41328, 339360, 28477440, 489877920, 7325176320, 501467630400, 14323336634880, 333439476289920, 21001701037363200, 849627551212876800, 27872303353627299840, 1742879646852427791360, 90170933394707691724800
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\3, (2*n+k)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/(2*n+1)!;

Formula

a(n) = (n!/(2*n+1)!) * Sum_{k=0..floor(n/3)} (2*n+k)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A376385 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x*log(1-x))^2 ).

Original entry on oeis.org

1, 0, 4, 6, 280, 1620, 67788, 844200, 36344992, 752867136, 34869857040, 1039132179360, 52776841318848, 2066262237673920, 115959403155851136, 5617102749187849920, 348802585405252070400, 20063354348482794961920, 1375625132090917881338880
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x*log(1-x))^2)/x))
    
  • PARI
    a(n) = 2*n!*sum(k=0, n\2, (2*n+k+1)!*abs(stirling(n-k, k, 1))/(n-k)!)/(2*n+2)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 + x*A(x) * log(1 - x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371230.
a(n) = (2 * n!/(2*n+2)!) * Sum_{k=0..floor(n/2)} (2*n+k+1)! * |Stirling1(n-k,k)|/(n-k)!.
Showing 1-7 of 7 results.